Problem: The product of two 2-digit numbers is $3774$. What is the smaller of the two numbers?
The prime factorization of 3774 is $3774=2\cdot3\cdot17\cdot37$.

$2$ and $3$ are troublesome here, because they are both 1-digit factors. We can deal with them by multiplying them by some other factor to produce a bigger factor.

One thing we might try is multiplying them together, but $2\cdot3=6$ is still only one digit.

If we try to put both of them with the $17$, that produces $2\cdot3\cdot17=102$, which is too many digits. Putting them with the $37$ would be even bigger, so that won't work either.

So we have to put one of them with each of the other factors. We can't put $3$ with $37$ because $3\cdot37=111>100$, so the only thing we can do is say $2\cdot37=74$ and $3\cdot17=51$. The smaller of those two is $\boxed{51}$.